Tight wavelet frames generated by the Walsh polynomials (Q2867966)

The author considers constructions of tight wavelet frames generated by the Walsh polynomials on the positive half-line \(\mathbb{R}^+\) using extension principles. Let \(p\) be a fixed natural number. Addition \(\oplus\) and subtraction \(\ominus\) on \(\mathbb{R}^+\) are defined modulo \(p\). Walsh polynomials are defined as linear combinations of so-called generalized Walsh functions \(w_m(x)\), \(m \in \mathbb{N}\cup \{0\}\).NEWLINENEWLINEThe main result (Theorem~3.1) is the following unitary extension principle for \(L^2(\mathbb{R}^+)\): Let \(n \in \mathbb{N}\), and let \(\varphi \in L^2(\mathbb{R}^+)\) be a compactly supported \(p\)-refinable function, that is, NEWLINE\[NEWLINE \varphi(x) = p \sum_{k=0}^{p^n-1}a_k \varphi(px \ominus k), NEWLINE\]NEWLINE generated by the Walsh polynomial \(m_0(\xi)=\sum_{k=0}^{p^n-1}a_k \overline{w_k(\xi)}\) with \(\varphi(0)= 1\). Then, for the combined \(p\)-MRA masks \(\{m_0 , m_1 , \dots , m_L \}\), where NEWLINE\[NEWLINE m_\ell(\xi)= \sum_{k=0}^{p^n-1}b_k^\ell \overline{w_k(\xi)}, NEWLINE\]NEWLINE the system \(\{\psi_{\ell,j,k}= p^{j/2} \psi^\ell(p^j x \ominus k), j \in \mathbb{Z}, k \in \mathbb{N}\cup\{0\},\ell = 1, 2,\dots , L\}\) forms a tight wavelet frame of \(L^2(\mathbb{R}^+)\) provided that the \(p \times (L+1)\) matrix \(M(\xi):=\left( m_\ell(\xi\oplus (i-1)/p )\right)_{i=1,\dots,p,\ell=0,\dots,L}\) satisfies \(M(\xi)M^\ast(\xi)=I_p\) for a.e. \(\xi \in \mathbb{R}^+\).